no. of Positive Divisor.

[man111]

(1) The number of positive divisors of [tex]17![/tex] is

[tex](A) 576
(B) 8568
(C) 8540
(D) 16![/tex]

(2) Find the no. of ordered pair [tex](x,y)[/tex] such that the LCM of [tex]x,y[/tex] is [tex]2^3 \times 3^4 \times 5^6[/tex]

[martin123456]

1
[tex]17!=2^{14}.3^{6}.5^3.7.11.13.17 \Rightarrow[/tex] the number of positive divisors is [tex](14+1)(6+1)(3+1)(1+1)^4=6720[/tex]
2
Since [tex]lcm(x,y)=2^3.3^4.5^6[/tex] is follows that [tex]x=2^{a_1}3^{b_1}5^{c_1}[/tex], [tex]y=2^{a_2}3^{b_2}5^{c_2}[/tex] and [tex]a_i,b_1,c_i \in \mathbb{N_0}[/tex] and [tex]\max{a_i}=3,\max{b_i}=4,\max{c_i}=6[/tex]. Exactly one of [tex]a_i[/tex] is fixed and so on. So we have [tex]2^3[/tex] different possibilities. For example let us fix [tex]a_1=3[/tex], [tex]b_1=4[/tex],[tex]c_1=6[/tex] and let [tex]a_1\le 2[/tex],[tex]b_2\le 3[/tex],[tex]c_2\le5[/tex]. Then we have [tex]2^3.3.4.6[/tex] possibilities. Now let us fix just one of [tex]x_1[/tex] to be its max value. So we have [tex]2^3.4.6+2^3.3.4+2^3.3.6[/tex] possibilities. Now fix two. We have [tex]2^3.6+2^3.3+2^3.4[/tex] possibilities. No fix all - we have [tex]2^3[/tex]. We have to sum these.